If you’re seriously underwater and headed to foreclosure, what would you say if the lender suddenly offered you the chance to remain in your home as a tenant for an extended period plus have your mortgage debt wiped away? Would you say yes?

Or would you instead conclude: Hey, why pay rent? It’s going to take the bank more than a year to complete the foreclosure and evict us, so why not just stay put and save some money?

One of the country’s largest banks is about to find out which choice significant numbers of distressed owners make in response to a new foreclosure-avoidance plan it calls “mortgage-to-lease.”

Bank of America is sending proposals later this month to upward of 1,000 customers in Arizona, New York and Nevada. If the reaction is positive, the program is likely to be expanded to other states, and it could become a model for the biggest players in the mortgage market, Fannie Mae and Freddie Mac.

Here’s how it works: The homeowners in the initial pilot tests have all previously been offered a variety of alternatives to foreclosure: loan modifications, forbearance on payments, short sales and “deeds-in-lieu,” where the borrower hands the property title back to the bank and moves out. But these homeowners either have not been able to qualify or have not responded to the bank’s proposals. They’re now essentially at the end of the line: There are no other standard lender remedies available to keep them out of foreclosure. In fact, most lenders are now speeding up the pace of foreclosures to get heavy loads of defaulted mortgages off their books.

In the mortgage-to-lease plan, a borrower is expected to deed back his home to the bank; in exchange, he gets multiyear lease terms at or slightly below the going market rent for the unit, provided he has monthly income to make the payments. When the lease expires, there is no guarantee that he will have a shot at repurchasing the home.

Bank of America officials acknowledge they are unsure how many homeowners will take them up on the offer. “This is an experiment — we just don’t know,” said Dave Steckel, the bank’s home retention strategy executive. It’s entirely possible, he concedes, that some owners will simply choose to proceed to foreclosure.

The plan has clear benefits for Bank of America. Since foreclosures generally trigger deeper losses to lenders than deeds-in-lieu of foreclosure, the bank gets back troubled properties at lower costs. Since it intends to sell those houses to investment groups as rentals with income-qualified tenants already living on the premises, the bank expects to obtain overall higher returns than it would by selling them as vacant, post-foreclosure units.

Though Bank of America has not discussed details of the plan with foreclosure-prevention and homeownership advocates, leaders of some groups had generally positive reactions to the concept.

“I’m glad to see the [lending] industry moving in this direction and away from their usual ‘Foreclosure; you’re gone; goodbye,’ ” said Colleen Hernandez, president and chief executive of the nonprofit Homeownership Preservation Foundation, which runs a hotline and counseling service that assisted 200,000 consumers last year. Not only might a mortgage-to-lease approach keep families in their homes, she said, but it should help neighborhoods by cutting down on the number of vacant, deteriorating properties that create blight and depress real estate values.

Hernandez’s main concerns with mortgage-to-lease concepts — she emphasized that she has not seen details of Bank of America’s version — are that they need to “offer an agreed-upon path back to homeownership” at the end of the lease period and that they should provide advance counseling for borrowers. Bank of America officials say their plan does not rule out repurchases but leaves that to the discretion of the investor.

David Berenbaum, chief program officer for the National Community Reinvestment Coalition, a nonprofit umbrella group, called mortgage-to-lease “a very compelling idea,” but only if the lender “has exhausted every other option” to keep people in their houses.

Where is this all headed and how fast? It’s uncertain at this stage, but if Bank of America’s pilot projects go well, with both consumers and investors enthusiastic, look for mortgage-to-lease to spread to other large lenders fast, potentially becoming a new tool in the national foreclosure-avoidance effort.